## 2 Design and Operation Principle of the Actuator  
The actuator is composed of a cylindrical frame and a piezoceramic ring glued on top of the cylinder (Figure 1). The frame is made from beryllium bronze C17200, while ring material is hard piezoceramic PIC181 (PI Ceramics, Lederhose, Germany). The spherical payload is placed on the top of the ring. The top electrode of the piezo ceramic ring is divided into six equal sections. The beryllium bronze cylinder has three cut-outs used to form three supports and to make six sectors along the circumference of the cylinder. The length of these sectors and their position corresponds with the length and position of the electrodes. The three supports are used to transfer vibrations of the piezo ceramic ring to the contact zone and to induce planar motion of the whole actuator on basis of vibro-impact principle. The other three sections are used to generate rotations of the spherical payload about three axes. It must be mentioned that there are an additional six cuts made in the cylinder used to separate six zones of the cylinder. These additional cuts reduce the vibration coupling of different zones of the actuator. In addition, alumina oxide spherical contacts are glued at the bottom of the supports and on the top of the piezo ceramic ring in each zone. Spherical contacts ensure higher friction force between contacting surfaces and allow to locate them on the antinodes of the particular vibration mode. The design of the actuator is shown in Figure 1, while Figure 2 and Table 1 represent geometrical characteristics. Figure 1: Design of the actuator: (**a**)—assembled view; (**b**)—exploited view; 1—beryllium bronze cylinder; 2—piezoceramic ring; 3—spherical contacts used for linear motion; 4—spherical contacts used for angular motion; 5—spherical payload. The piezoelectric actuator has a simple design and can be scaled. The introduced actuator occupies a 2.99 cm\({}^{3}\) space together with the spherical payload. The total mass of the actuator without payload is 12.55 g.  
The operation principle of the actuator is based on the excitation of the two vibration modes, i.e., the first radial vibration mode of the supports and the third radial vibration mode of the piezo ceramic ring. A single harmonic signal is used to excite a particular motion of the actuator. If one of the Planar\({}_{1}\)-Planar\({}_{2}\) electrodes (Figure 2) located above the support is excited by harmonic signal at the resonant frequency of the first radial mode of the support, then the linear motion of the actuator is induced into the normal direction of the support. Reverse linear motion of the actuator is obtained when the other two Planar\({}_{1}\)-Planar\({}_{2}\) electrodes are excited by harmonic signal with the same frequency. In order to generate angular motion of spherical payload, one of the sections dedicated to controlling angular motion (Rotary\({}_{1}\)-Rotary\({}_{3}\)) should be affected by harmonic signal with frequency equal to the natural frequency of the third radial mode of piezo ceramic ring. Reverse motion is obtained by exciting two opposite electrodes. All three Rotary\({}_{1}\)-Rotary\({}_{3}\) segments must be affected by three harmonic signals with a phase difference of 120\({}^{\circ}\) to obtain rotation of the payload about the \(Z\)-axis. It should be noted that preload of the actuator induced by spherical payload affects vibration amplitudes of the actuator. Therefore, the difference between amplitudes of the active and passive zones of the actuator is increased. This allows reducing disturbance in the motion direction when the particular electrode is affected. Motion control of the actuator was implemented using two harmonic signals with different frequencies that are controlled by two independent digitally controlled switch boxes (Table 2). The planar motion of the actuator or rotary motion of the payload is excited by switching on or off corresponding switches.