# SiMFlex Micromanipulation Cell with Modular Structure
## 3 Manipulation Module  
In Figure 2, the micro manipulating module is presented. It has a basic compliant gripper part (1) and a piezoelectric stack actuator (2). The jaws (3) can be attached in two different ways: one with interchangeable jaws (3a) and the other with bending piezoelectric jaws (3b). For interchangeable jaws, three variants are proposed: circular, rectangular, and general shapes. The bending piezoelectric jaws can be closed at the same time or only one at a time. The basic compliant gripper part (1) possesses a symmetry line along the piezoelectric actuator and it has 12 identically flexible joints with a corner filleted shape (Figure 2b), defined by the length of the flexure hinge (l), the thickness (t), and radius (r). When the radius r increases, a less stress concentration and displacement results at the flexure hinge, but it increases the transmitted force [2]. The movement of the flexure hinge in the mechanism with various cross-section used in the gripper structure can be analyzed with the generic equations of Castigliano theorem. \[ \begin{array}{l}\theta_{\mathrm{i}}=\int\limits_{0}^{\mathrm{I_{f}}}\frac{ \mathrm{M_{i}}}{\mathrm{E_{i}}}\frac{\partial\mathrm{M_{i}}}{\partial\mathrm{ M}}\mathrm{d}\mathrm{x}\\ y_{\mathrm{i}}=\int\limits_{0}^{\mathrm{I_{f}}}\frac{\mathrm{M_{i}}}{\mathrm{ E_{i}}}\frac{\partial\mathrm{M_{i}}}{\partial\mathrm{F}}\mathrm{d}\mathrm{x}\end{array}  \tag{1}\]  
where: M\({}_{i}\)--bending moment, I--inertia moment, E--[PERSON]'s modulus of the flexure hinge. Figure 1: Micromanipulator System with Functional Flexibility (SiMFlex) cell (**a**) increase precision positioning y axis and (**b**) increase positioning x axis. The displacement of flexure hinge is directly proportional to the length of the flexure hinge (l\({}_{\text{f}}\)). As length increases the stiffness decreases and a displacement increase occurs at the same time. The analysis also can indicate that the overall thickness (b) and the flexure hinge thickness (t) determine the bending stiffness of the flexure hinge. The increase of these parameters increases the bending stiffness and hence reduces the output displacement. The mechanical advantage for main compliant gripper can be found as the ratio of the input displacement from stack piezoelectric actuator, \(u_{Ax}\), to the output displacement from jaws, \(u_{Hy}\) (Figure 3). The input displacement is determined by applying only the force Pi at the input port by using, again, [PERSON]'s theorem in the form:  
\[u_{Ax}=\frac{\partial U}{\partial Pi} \tag{2}\]  
The displacement at the output port under the action of Pi can be calculated by using a dummy force Po through:  
\[u_{Hy}=\frac{\partial U}{\partial Pi} \tag{3}\]  
The methodology for relations between displacement and force applied to the micro-objects was detailed by authors in [2; 15]. Figure 3: FEM analysis for compliant gripper: (**a**) displacement and (**b**) Von Mises stress. Figure 2: Manipulation module: (**a**) interchangeable jaws and (**b**) bending piezoelectric jaws. By combining Equations (2) and (3), the mechanical advantage is obtained in the form:  
\[Amplification=\frac{\Delta y}{\Delta x}=\frac{m_{mc}}{m_{act}} \tag{4}\]  
The Finite Element Method (FEM) analysis for compliant gripper was realized, using Solidworks Simulation, considering a maximum displacement of 10 \(\upmu\)m for the piezoelectric stack actuator. The basic compliant gripper part has overall dimensions of 54 \(\times\) 41 \(\times\) 3.80 mm, also, the flexure hinges have the dimensions: \(\mathrm{l}_{f}\) = 2 mm, t = 0.56 mm, and r = 0.3 mm. The mechanism out-of-plane thickness was w = 3.80 mm. The material used was steel alloy with an elastic modulus E = \(2.1\cdot 10^{11}\) N/m\({}^{2}\). and [PERSON]'s ratio \(\upmu\) = 0.28. The simulation considered the output point is free, condition for which the displacements (Figure 3a), and the Von Mises stress (Figure 3b) have been evaluated by means of the resulting input and output displacements.