## 2 Design of the Micromanipulation Device Using Piezoelectric Actuators
### Gripper Based on Parallel Arrange of Piezoelectric Actuators  
The proposed gripping system is constructed using a self-designed support structure that carries a pair of piezoelectric actuators in a parallel arrangement. The holding support device uses an additive manufacturing strategy based on the 3D printing technique. This strategy reduces the weight of the gripping support. The selected actuators attached to the support are bimorph-shaped devices that can bend unilaterally or bilaterally. Each actuator is controlled using a voltage source ranging from 0 to 150 volts, which is controlled using a pulse-width modulation strategy (Figure 2).  
The prefabricated piezoelectric actuators were obtained from Thor Labs Company, and have the following characteristics: a displacement range of 450 microns, a hysteresis effect larger than 15%, and a curie temperature of 230 \({}^{\circ}\)C. Although in this study the actuator motion is usually ruled by a single oscillating mode [19; 20; 21], this is commonly used as a simplified form of mathematical representation, typically presented as a partial differential equation. This simplified form satisfies an ordinary differential equation based on applying a spatial discretization method, which is obtained using [PERSON]'s method [22; 23; 24]. Based on this technique, the temporal-spatial motion of the actuator, \(u_{z}(x,t)\), can be represented as \(u_{z}(x,t)=\varrho_{t}\phi(x)\). Here, \(\varrho_{t}=u_{z}(L,t)\) represents the deflection of the actuator's distal element with respect to its resting position, while \(\phi(x)\) is a unitary normalized function, that is, \(\phi(x)=1\).  
Figure 1: Description of the robotic arm and the gripping device based on bimorph piezoelectric actuators.  
The dynamics of the piezoelectric actuator are defined by the following two differential equations that aggregate the actuator's mechanical section and the device's electrical dependence. The proposed mechanical equation integrates the effect of the acceleration related to the end-effector of the robotic arm.  
\[\begin{gathered} m\ddot{x}+\big{[}b_{1}xsign(x)+b_{2}x^{2}\big{]} sign(\dot{x})+\\ \ddot{R}_{1}x+\ddot{R}_{2}x^{2}sign(x)-\{\dot{\Sigma}_{1}+\ddot{ \Sigma}_{2}xsign(x)\}V=-\ddot{m}\ddot{r}\\ C\dot{V}+I+\{\dot{\Sigma}_{1}+\dot{\psi}xsign(x)\}\dot{x}=0\end{gathered} \tag{3}\]  
In expression (3), the dynamic actuation mode for the actuator is represented by coupled mechanical-electrical forms. In this lumped parameters model, the mechanical dynamics characterized by the lumped displacement, \(x\), are driven by the voltage signal, \(V\), which turns out to be regulated by the input current, \(I=VR^{-1}\), based on [PERSON]'s law. This current is supplied by the external source, which is the actual controller of the system under study. The parameter \(m\) represents the mass of the actuator, \(b_{1}\) is the parameter related to the motion direction corresponding to the hysteresis effect depending on the motion, \(b_{2}\) represents the hysteresis effect concerning the motion displacement and velocity, and \(\ddot{K}_{1}\) and \(\ddot{K}_{2}\) are the constants defining the dependence of \(x\) and \(x|x|\), respectively, on the motion dynamics. Similarly, \(\dot{\Sigma}_{1}\) and \(\dot{\Sigma}_{2}\) represent the time-dependence including the internal voltage relationship. Considering that the gripper is attached to the distal section of the robotic arm, the mechanical effect is taken into consideration, here with the acceleration enforced by this interconnected motion, \(\dot{m}\ddot{r}\), where \(\dot{m}\) is the mass of the last link in the arm and \(\ddot{r}\) is the acceleration of the same section. The parameter \(C\) represents the capacitance of the sections making up the bimorph configuration, while \(\ddot{\psi}\) is the back mechanical-electrical section.