# Single Cylinder-Type Piezoelectric Actuator with Two Active Kinematic Pairs
## 3 Mathematical Model of the Contact Force Control
Then orientation of forces \(\overrightarrow{f}_{3,4,5}\) of the sphere can be calculated by formula [15]:  
\[\left[\overrightarrow{f}_{J}(\alpha_{I},\beta_{I},\gamma_{I})\right]=\cos^{ -1} \begin{bmatrix}\frac{\sqrt{R^{2}-D^{2}}}{R}&0&\sqrt{D^{2}-R^{2}+1}\\ -\frac{\sqrt{R^{2}-D^{2}}}{2R}&\frac{\sqrt{3}}{2}\frac{\sqrt{R^{2}-D^{2}}}{R} &\sqrt{D^{2}-R^{2}+1}\\ -\frac{\sqrt{R^{2}-D^{2}}}{2R}&\cos\left(180^{\circ}-\cos^{-1}\left(\frac{ \sqrt{3}}{2}\frac{\sqrt{R^{2}-D^{2}}}{R}\right)\right)&\sqrt{D^{2}-R^{2}+1} \end{bmatrix}  \tag{4}\]  
To choose an appropriate electrode segment for excitation, it is necessary to know the total force \(\overrightarrow{F}\) of all segments:  
\[\overrightarrow{F}^{C}(t)=\sum_{I=6}^{n}\overrightarrow{f}_{J}(t),\,\text{ in case }J=6,7,8\ n=3; \tag{5}\] \[\overrightarrow{F}^{C}(\tau)=\sum_{J=3}^{n}\overrightarrow{f}_{J }(\tau),\,\text{in case }J=3,4,5\ n=3;\]  
The electrode segment that is the nearest to the total force \(\overrightarrow{F}\) vector must be selected for excitation and its numeric value must be equal to maximum \(f_{max}\) (Figure 8a). To obtain precision of the motion, a force that is cardinally in the opposite orientation than the total force must be activated. Its value must be equal to the minimum value of the force \(f_{min}\). The value of the third electrode is calculated after solving a system of equations [14]:  
Figure 8: (**a**) Trajectory of the cylinder; (**b**) Sphere rotation trajectory. \[\left\{\left| \begin{array}{c}\overrightarrow{F}\\ \overrightarrow{F}\end{array} \right|\cos\mu=\sum_{J}^{n}\left|\overrightarrow{f}_ {J}\right|\cos\phi_{J} \tag{6}\]  
\[\text{where}\left|\overrightarrow{F}\right|=\tan^{-1}\frac{T_{y}}{T_{x}};\phi_ {J}=\overrightarrow{F}\overrightarrow{F}_{J}\text{ and }\left|\overrightarrow{f}\right|= \begin{cases}f_{ min},\text{ if }\left|\overrightarrow{f}_{J}\right|=0,\\ f_{max},\text{ if }\phi_{J}=min(\left[\phi_{J}\right])\\ \left|\overrightarrow{f}_{J}^{*}\right|\end{cases} \]  
The integration scheme of the fine trajectory planning algorithm for one active element with two kinematic pairs, when parallel calculations are used, is presented in Figure 9.  
The sphere of the radius \(R=20\) mm and the cylinder of the radius \(D=16.5\) mm were selected for the numerical experiment. The trajectory of the sphere rotation around different axes was described by formulas (Figure 10):  
\[S(\tau)= \begin{cases}\theta(\tau)=4\tau^{3}+10\\ \phi(\tau)=15\tau^{2}+2\tau\end{cases} ,\text{ where }t\in[0,3] \tag{7}\]  
\[C(t)= \begin{cases}x(t)=t\cos(t)\\ y(t)=2t\sin(t)\end{cases} ,\text{ where }t\in[0,3] \tag{8}\]  
Figure 9: The integration system of the fine trajectory planning algorithm. The trajectory of the cylinder is given in Figure 11. The forces of the sphere must be in the interval \(\left|\overrightarrow{f}_{6,7,8}\right|\in[0.5,1]\) mN, and the forces of the cylinder in the interval \(\left|\overrightarrow{f}_{3,4,5}\right|\in[3,5]\) mN. The results of the calculation of the cylinder forces and the sphere electrode segments are presented in Figure 12a,b. The numerical experiment confirmed that the fine trajectory planning algorithm is successfully integrated and is suitable for both movements: the movement on the plane and sphere rotation on certain points of the contact. This means that kinematic pairs of the actuator can be managed separately. Figure 11: The cylinder motion trajectory. Figure 10: The sphere rotation trajectory (in dots).