# Study and Simulation of an Under-Actuated Smart Surface for Material Flow Handling
## 3 Analytical Model
### Friction Forces Model  
In this subsection, considering the previous assumptions regarding sliding and pure rolling and taking into account that the friction forces oppose motion, the directions of the two forces (\(F_{par}\), \(F_{perp}\)) and their final formulation are determined. Thanks to the third assumption, these directions are related only to the object velocity at the contact point with the rotor, \(\overrightarrow{V_{p}}=[\dot{x}_{p},\dot{y}_{p}]\); an example is shown in Figure 4a. In particular, the velocity \(\overrightarrow{V_{p}}\) can be split into a perpendicular and a parallel component (Figure 4a), and the two forces \(F_{par}\) and \(F_{perp}\) are in the opposite directions of these velocities. In Figure 4a, three more angles are introduced: \(\varepsilon\), \(\beta\), and \(\alpha\) (all angles are defined positive in the anti-clockwise direction from the \(x\) axis):  
* \(\varepsilon\) is the angle between the rotor axis and the \(x\) axis;
* \(\beta\) is the angle between the velocity of the object's contact point and the \(x\) axis;
* \(\alpha\) is obtained by the difference between the previous two (\(\alpha=\varepsilon-\beta\)), so it is the angle between the velocity of the body's contact point and the rotor axis. Figure 4: Schematic representations of: (**a**) velocities and forces exchanged in a contact point, (**b**) the calculation of the velocity in the contact point, (**c**) the resulting forces from the contacts. The velocity of the object's contact point is determined using the formula of the kinematics of a rigid body: \(\overrightarrow{V_{p}}=[\dot{x_{g}}\hat{\tilde{i}},\dot{y_{g}}\hat{\tilde{y}}]+ \dot{\theta}_{g}\hat{k}\ \times\ \overrightarrow{\widetilde{G}P}\), where \([\dot{x}_{g},\dot{y}_{g}]\) and \(\dot{\theta}_{g}\) are the linear and angular velocity of the object, and \(\overrightarrow{\widetilde{G}P}\) is the vector that connects \(\widetilde{G}\) to the contact point \(P\). The velocity identification is also reported in Figure 4b, where the piece is schematized with the shape of the base, and the rotors with a grid of points. This scheme presented in Figure 4b is also useful in visualizing how the rotor-object contact points are determined. In fact, knowing the values \([x_{g},y_{g},\theta_{g}]\) and the mathematical formulation of the base area with respect to these, it is possible to know the set of coordinates of the points inside the base perimeter. More specifically, given a planar reference frame \(\{\widetilde{G},\widetilde{x},\widetilde{y}\}\) with coordinates \(\widetilde{x},\widetilde{y}\), centered in \([x_{g},y_{g}]\) (\(\widetilde{G}\)) (Figure 4b) and oriented by \(\theta_{g}\), let \(Q\) be a set of geometrical parameters that depends on the shape of the object base (e.g., radius for a circle or base and height for a rectangle) and let \(GD\in Q\). Let \(\Lambda\) be a set of indexes. For every \(\lambda\in\Lambda\), let \(F_{\lambda}:\mathbb{R}^{2}\ \times\ Q\rightarrow\mathbb{R}\) be functions such that \(W=\{(x,y):F_{\lambda}(x,y,GD)=0,\forall\lambda\in\Lambda\}\) is the perimeter of the object base. The area of the object base, \(D\), is defined as \(D=\{(\widetilde{x},\widetilde{y}):F_{\lambda}(\widetilde{x},\widetilde{y}, GD)\leq 0,\forall\lambda\in\Lambda\}\). Therefore, given the \(x_{R},y_{R}\) coordinate pairs that define the position of the rotors in the array (which are fixed), it is verified whether or not they belong to the set, i.e., if \( \begin{bmatrix}x_{R}\\ y_{R}\end{bmatrix} = \begin{bmatrix}x_{g}\\ y_{g}\end{bmatrix} +R_{\theta_{g}} \begin{bmatrix}\widetilde{x}\\ \widetilde{y}\end{bmatrix} \) has solution for a pair of \(\widetilde{x},\widetilde{y}\in D\), where \(R_{\theta_{g}}= \begin{bmatrix}\cos\theta_{g}&-\sin\theta_{g}\\ \sin\theta_{g}&\cos\theta_{g}\end{bmatrix} \) is the rotation matrix of the reference frame with coordinates \(\widetilde{x},\widetilde{y}\) and centered in \(\widetilde{G}\).