## 2 Material and Methods  
### Vacuum Microgripper and Its Working Principle  
Classic vacuum microgrippers consist of two parts: a hollow body and a thin cannula (Figure 1). The three manipulation tasks (grasping, handling, releasing) are carried out through a simple working principle. When a pressure difference, \(\Delta p\), between the inflow section and the volume inside the gripper is created by a vacuum pump connected to the gripper, airflow is forced through the cannula and inside the device, generating a suction force that picks up the object (grasping phase) (Figure 1b). When the object sticks against the cannula end (handling phase) (Figure 1c), the gripper must provide a continuous handling force (\(F_{H}\)) that, according to Equation (1), must be sufficiently high to overcome the weight force (\(mg\)) minus the adhesion forces (\(F_{AD}\)):  
\[F_{H}=\Delta pA_{0}\geq mg-\ F_{AD} \tag{1}\]  
where \(A_{0}\) is the cannula's cross-sectional area, \(m\) the object mass and \(g\) the gravitational acceleration. The releasing phase begins when \(\Delta p=0\). However, the weight force is not often enough to strip off the object due to high adhesion forces (i.e., Van der Waals, electrostatic, surface tension) acting between the grasped object and the cannula's surface (Figure 1d). Figure 1: Off (**a**), grasping (**b**), handling (**c**) and releasing (**d**) manipulation phases of a classic vacuum microgripper. When the objects to manipulate are very small in size (less than 10 \(\upmu\)m), the importance of adhesion forces (proportional to the object surface) arises and can be dominant over gravitational and inertial forces (proportional to the object volume) (Figure 2). In order to overcome this issue, some authors [12] patented a different kind of vacuum microgripper, equipped with an automatic release tool. A releasing mass (in red in Figure 3), combined with a thin needle, is inserted inside the gripper body and can move up and down in the axial direction. A mechanical stop (not shown in figure) limits the vertical displacement of the releasing mass. In addition, two lateral holes are provided on the lateral gripper body surface to enhance airflow inside the device and to allow the releasing mass to be lifted also during the handling phase. The aerodynamic force (\(F_{M}\)) acting on the releasing mass during the device operation (grasping and handling phase) (Figure 3b,c) is the sum of two contributions, according to the following equation:  
\[F_{M}=F_{P}+F_{V} \tag{2}\]  
Figure 3: Off (a), grasping (b), handling (c) and releasing (d) manipulation phases of the studied vacuum microgripper. Forces acting on the grasped object and the releasing mass during the handling phase (e). Figure 2: Orders of magnitude of gravitational, electrostatic, Van der Waals and tension forces vs. object size [13]. The pressure force (\(F_{P}\)) is generated by the difference between the average pressure acting on the upstream (\(p_{xx}\)) and the downstream (\(p_{dx}\)) base of the releasing mass (\(A_{c}\)) and is defined as  
\[F_{P}\cong(p_{xx}-p_{dx})A_{c} \tag{3}\]  
The viscous force (\(F_{V}\)) is generated by shear stresses (\(\tau_{w}\)) acting on the releasing mass lateral surface (\(A_{L}\)) in the annulus region, and is defined as  
\[F_{V}=\int_{A_{L}}\tau_{w}dA \tag{4}\]  
When the pressure difference is removed, i.e., \(\Delta p\) = 0, the releasing mass moves downward and helps with object detachment (see Figure 3d). As shown in [14], this integrated release tool significantly improves the device's release effectiveness. Greater values of the releasing tool mass (\(M\)) allow one to overcome high adhesive forces, especially for small objects, according to the following equation:  
\[(m+M)g>\ F_{AD} \tag{5}\]  
The value of \(M\) must be carefully chosen (see Figure 3e) as follows:  
\[M\ \leq\ \frac{F_{M}}{g} \tag{6}\]  
to ensure that \(F_{M}\) is sufficiently high to correctly lift the releasing tool and guarantee proper device operation.