**5. Conclusions**

A new piezoelectric actuator with two active kinematic pairs was proposed and investigated. A novel electrode configuration scheme was used to excite vibrations of the actuator. Besides, a mathematical model of the contact force control and a trajectory planning algorithm were proposed and evaluated. Exciting different electrodes and controlling the contact force in kinematic pairs with brakes can create rotation of the sphere or movement on the plane. The trajectory planning method was validated by the numerical analysis. A prototype actuator was made, and mechanical and electrical characteristics were measured. The results of the numerical simulation and experimental measurements were in a good agreement.

**Supplementary Materials:** The following are available online at http://www.mdpi.com/2072-666X/9/11/597/ s1. Video S1: A motion of the sphere when driving voltage is applied to the actuator.

**Author Contributions:** J.J.-B. and D.M. were responsible for FEM analysis; A.D. and G.K. designed the trajectory planning algorithm; R.B. conceived the idea, supervised the experiments, and analyzed the data; V.J. produced the prototype and performed experimental measurements. All the authors contributed to the manuscript preparation.

**Funding:** This research was funded by European Regional Development Fund, gran<sup>t</sup> number DOTSUT-234.

**Acknowledgments:** This research was funded by the European Regional Development Fund according to the supported activity No. 01.2.2-LMT-K-718 under the project No. DOTSUT-234.

**Conflicts of Interest:** The authors declare no conflict of interest.
