# Triangular-Shaped 5-DOF Piezoelectric Robot for Optical Lens Positioning
## 3 Numerical Investigation of Robot  
A numerical investigation of the robot was performed to analyze the electrical and mechanical characteristics as well as vibrations of the contacts and coupling ratios between vibrations. A numerical model was built using Comsol 5.4 software. The geometric parameters of the robot are shown in Figure 2. Material characteristics were defined as follows: DIN 1.4301 stainless steel was used for the passive triangle layer, the piezoceramic PIC 181 (PI Ceramics) was used for the ring, and finally, alumina oxide material properties were used for semispherical contacts. The material properties used to build the model are given in Table 2. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Motion Direction**} & \multicolumn{4}{c}{**Angular Motion**} & \multicolumn{4}{c}{**Planar Motion**} \\ \cline{2-7}  & **SW\({}_{1}\)** & **SW\({}_{4}\)** & **SW\({}_{4}\)** & **SW\({}_{4}\)** & **SW\({}_{5}\)** & **SW\({}_{6}\)** \\ \hline
0\({}^{\circ}\) & 1 & 1 & 1 & 1 & 0 & 0 \\
120\({}^{\circ}\) & 0 & 0 & 0 & 0 & 1 & 0 \\
240\({}^{\circ}\) & 0 & 0 & 0 & 0 & 0 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Switching between electrodes used to control motion direction. Figure 3: Excitation schematics of the robot; 1—excitation signal source with two independent channels; 2—switch box for planar motion control; 3—switch box for angular motion control. The payload was simulated as a distributed mass placed on the contacts located on top of the piezoceramic ring. The gravity force of the robot was included in the model. Electric boundary conditions were defined depending on the study and are given below in this section. The excitation schematic of the robot is shown in Figure 3.  
Firstly, a modal analysis of the robot was performed. The goal of the study was to indicate vibration modes and natural frequencies suitable for robot operation. The electrical boundary condition of the piezoceramic ring was set to obtain a short circuit status. Modal shapes of the actuator are shown in Figure 4.  
Figure 4a shows that the first bending mode of trapezoidal cantilevers occurs at the frequency of 13.79 kHz. It can be seen that the highest displacement amplitudes are obtained at the ends of the cantilevers, while contacts used for angular motion have very small displacements. Therefore, forced vibrations of trapezoidal cantilevers will have a minor influence on the vibration of contacts located on the top of the ring. The third radial vibration mode of the piezo ceramic ring is obtained at a frequency of 95.75 kHz. Analyzing this mode, it can be seen that radial vibrations of the piezo ceramic ring have a minor influence on vibrations of trapezoidal cantilevers. Therefore, the angular position of the payload will not generate planar motion of the robot. Impedance and phase-frequency characteristics of the robot were analyzed in the frequency ranges of 13.78-13.88 kHz and 95.0-96.5 kHz. Numerical simulation was made when R\({}_{\text{0 deg}}\) and L\({}_{\text{0 deg}}\) segments were excited by the voltage of 50 V\({}_{\text{p-p}}\), separately, while the remaining segments of the piezo ceramic ring were set to open-circuit condition. The results of the calculations are shown in Figure 5.