# Motion Characteristics of Self-Sensing Piezoelectric Actuator for Yarn Micro-Gripper
## 2 Structure and Modeling of the Self-Sensing Micro-Gripper  
### The Structure of the Self-Sensing Micro-Gripper  
In order to achieve the required clamping range and clamping force for yarn clamping, the self-sensing clamping device investigated in this paper is driven by a cantilever beam structure. In this paper, we chose the bimorph piezoelectric bending actuator (BPBA) composed of two layers of PZT-5H piezoceramics and one layer of carbon fiber substrate. The physical diagram of the BPBA is shown in Figure 1. The bimorph piezoelectric ceramic can improve its performance, while the substrate ensures that the actuator can meet the requirements of long-term high-intensity work and reduce the destruction rate. Two layers of piezoelectric ceramics are bonded to the upper and lower surfaces of the substrate, which alternately expand and contract under a certain period of varying voltage, generating a bending moment and then the free end of the reciprocating bending motion. The clamping scheme is shown in Figure 2. The specific dimensions of the piezoelectric cantilever and the performance parameters of the materials used are given in Tables 2 and 3.  
operation of the BPBA, it was modelled and analyzed. As the fingers of the gripper are symmetrically distributed, only one of them needs to be studied. Figure 3c,d shows the three-view schematic diagrams of the actuator. The 1, 2, and 3 axes of the graph represent the X, Y, and Z axes, respectively, where \(L\) is the length of the actuator and \(W\) is the width of the driver. \(s_{u}\), \(s_{c}\) and \(s_{l}\) are the thicknesses of the upper piezoceramic layer, the middle carbon fiber layer, and the lower piezoceramic layer, respectively, and \(s_{u}=s_{l}=s_{p}\). A voltage is applied to the piezoelectric ceramic element and the driver is deformed by the inverse piezoelectric effect. According to the simplified piezoelectric equations of the first type:  
\[\left\{ \begin{aligned} & x_{1}=s_{11}^{E}X_{1}+d_{31}E_{3}\\ & D_{3}=d_{31}X_{1}+e_{11}^{E}E_{3}\end{aligned} \right., \tag{1}\]  
when piezoelectric ceramics are used as actuators, the strain \(x_{1}\) of the piezoelectric crystal in the 1-axis direction under the action of stress \(X_{1}\) and electric field \(E_{3}\) is  
\[\left\{ \begin{aligned} & x_{1}^{U}=s_{11}^{E}X_{1}^{U}-d_{31}E_{3}^{U}\\ & x_{1}^{L}=s_{11}^{E}X_{1}^{L}+d_{31}E_{3}^{L}\end{aligned} \right. \tag{2}\]  
At the same time, there is no piezoelectric effect in the middle layer. Therefore, the strain-stress relationship for each layer of the three-layer structure of the driver is  
\[\left\{ \begin{aligned} & X_{1}^{U}=(s_{11}^{E})^{-1}x_{1}^{U}+(s_{11}^{E})^{- 1}d_{31}E_{3}^{U}\\ & X_{1}^{C}=(s_{11}^{E})^{-1}x_{1}^{C}\\ & X_{1}^{L}=(s_{11}^{E})^{-1}x_{1}^{L}-(s_{11}^{E})^{-1}d_{31}E_{3 }^{L}\end{aligned} \right., \tag{3}\]  
where \(\left(s_{11}^{E}\right)^{-1}\) and \(\left(s_{11}\right)^{-1}\) are the Young's modulus of the piezoelectric layer and the substrate, respectively, which can be written as \(E_{p}\) and \(E_{c}\). Due to the symmetrical structure and opposite polarization directions of the upper and lower piezoelectric layers, pure bending of the mid-plane of the BPBA occurs at the driving voltage. Its bending curvature \(k\) can be expressed as  
\[k=\frac{d^{2}w}{dx^{2}}, \tag{4}\]  
where \(w\) is the deformation of the piezoelectric cantilever in the 3-axis direction. Then, the tensile strain of the piezoelectric cantilever in the 1-axis direction is  
\[x_{1}=kz_{1} \tag{5}\]  
Figure 3: Schematic view of the gripper and the BPBA. (**a**) View of gripper (open), (**b**) view of gripper (closed), (**c**) view on the 13-plane, (**d**) view on the 23-plane, (**e**) detail of the end of the BPBA after bending. Substituting into Equations (3) and (5)  
\[\left\{ \begin{array}{l}X_{1}^{U}=E_{p}kz_{1}^{U}+E_{p}d_{3}1E_{3}^{U}\\ X_{1}^{C}=E_{c}kz_{1}^{C}\\ X_{1}^{I}=E_{p}kz_{1}^{I}-E_{p}d_{3}1E_{3}^{L}\end{array}