## 2 Design and Working Principles  
The proposed piezoelectric USM consists of two ring-shaped piezoelectric transducers glued on the thin metal flanges that have disc shape (Figure 1). The spherical rotor is placed in between piezoelectric transducers. Transducers are driven by a harmonic electric signal with the frequency of higher vibration modes of the unimorph structure. The stator of the USM is a composite structure, in which the flanges and piezoelectric transducers are bonded using epoxy resin and are forming two unimorph piezoelectric actuators. This type of actuators has the benefit of generating larger flexural displacements with higher mechanical force and torque, which rotates a sphere. The proposed USM potentially can be used in optomechanical systems, humanoid robotics, military, and space applications [12]. \begin{table}
\begin{tabular}{c c c c c} \hline
**Authors** & **Operation Frequency [kHz]** & **Excitation Voltage (V\({}_{\text{p-p}}\))** & **Rotation Speed (RFM)** & **Size of the Motor [mm]** \\ \hline [PERSON] and et al. [4] & 93.6 & 40 & \(4.05\times 10^{-4}\) & \(33\times 20\) \\ [PERSON] and et al. [5] & 0.05-1.184 & 100 & - & \(-100\times-120\times-60\) \\ [PERSON] and et al. [6] & - & - & - & \(35\times 7\times 5\) \\ [PERSON] and et al. [8] & -41 & 200 & 100 & - \\ [PERSON] and et al. [9] & 64 & 120 & 32 & R = 12 \\ [PERSON] and et al. [10] & 57.3 & 40/60 & - & - \\ Proposed motor & 92.48/93.72 & 50 & 27/30 & \(36\times 28\) \\ \hline \end{tabular}
\end{table} 
Table 1: Comparison of the parameters of reviewed and proposed motors. Figure 1: _Cont._The basic design idea and structure of the developed USM is presented in Figure 1a. A spherical rotor (1) is inserted among two unimorph piezoelectric actuators, as presented in Figure 1b. The motor is symmetrical; therefore, the top and bottom actuators are identical. Piezoelectric ring-shaped transducers are made from hard type PZT-4 ceramics. The piezoceramic rings were polarized in the thickness direction, parallel with Z axis, and electrodes cover the upper and lower surfaces of the piezoelectric transducer. Electrodes on the upper surface are partitioned in 3 sections with an angle of 120\({}^{\circ}\) between them. The electrode on the bottom surface is continuous and glued on the flange. Three contact elements (4) are glued on the upper electrode (2) and they are located in the center of each segmented electrode. When the electric signal makes the piezoelectric transducer vibrate, then the frictional force acting between the contact element and the spherical rotor is generated, and the rotation of the rotor is obtained. Three bolts (5, 6) are used to get the motor assembled. The preload force between contact elements (4) and spherical rotor (1) can be adjusted by turning the nuts (5). The USM driving scheme and layout of segmented electrodes of the piezoelectric transducer are shown in Figure 1c. The setup consists of a three-channel signal generator (7) and controller (8). Three pairs of segmented electrodes of the upper and lower transducers are electrically connected to channels A, B, C. The harmonic electric signal is amplified by controller. Only one channel is active at a certain moment and it is used to rotate the sphere. Such setup allows achieving excitation of a specific segment of the piezoelectric actuator and induce movements in elliptical trajectories of the contact element (4). The operating principle of the proposed USM is provided by the elliptical motion of the excited contact elements (Figure 1b) when a higher bending mode of the actuator appears. While the driven electric signal is defined at particular frequency and applied for specific electrodes of piezoelectric actuator, the resonant vibrations are excited. Accordingly, the elliptical motion of the contact elements is obtained, and an oblique impact on the surface of the sphere is accomplished. As a result, the directional rotation of the rotor around one particular axis is generated.